how to prove that (3d_3 +4d_2 +3d_1 )^2 ≤5(d_1 ^2 +d_2 ^2 +d_3 ^2 +(d_2 +d_1 )^2 +(d_3 +d_2 )^2 +(d_1 +d_2 +d


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Question Number 201477 by York12 last updated on 07/Dec/23

$how to prove that$ ${({3 d}_{3} + {4 d}_{2} + {3 d}_{1})}^{2} ⩽ 5 (d_{1}^{2} + d_{2}^{2} + d_{3}^{2} + {(d_{2} + d_{1})}^{2} + {(d_{3} + d_{2})}^{2} + {(d_{1} + d_{2} + d_{3})}^{2})$
	Answered by deleteduser1 last updated on 07/Dec/23

	$L e t d_{1} = a; d_{2} = b; d_{3} = c$ $5 (3 a^{2} + 4 b^{2} + 3 c^{2} + 4 a b + 4 b c + 2 c a) \overset{?}{⩾}$ $9 a^{2} + 16 b^{2} + 9 c^{2} + 24 b c + 24 a b + 18 a c$ $\Leftrightarrow 3 a^{2} + 2 b^{2} + 3 c^{2} ⩾ 2 a b + 2 b c + 4 a c$ $a^{2} + b^{2} ⩾ 2 a b; 2 a^{2} + 2 c^{2} ⩾ 4 a c; b^{2} + c^{2} ⩾ 2 b c$ $\Rightarrow 3 a^{2} + 2 b^{2} + c^{2} ⩾ 2 a b + 2 b c + 4 a c \Rightarrow o r i g i n a l i n e q u a l i t y$ $i s t r u e .$
	Commented by York12 last updated on 07/Dec/23

	$thanks sir$
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